express the vector v = <-11 , 9> as a linear combination of x = <-2, -3> and y = <5, 5>

We want to express the vector \( \mathbf{v} = \langle -11, 9 \rangle \) as a linear combination of the vectors \( \mathbf{x} = \langle -2, -3 \rangle \) and \( \mathbf{y} = \langle 5, 5 \rangle \). This means finding scalars \( a \) and \( b \) such that:

\[
\mathbf{v} = a\mathbf{x} + b\mathbf{y}
\]

In component form, this becomes the system of equations:

\[
-11 = a(-2) + b(5)
\]
\[
9 = a(-3) + b(5)
\]

### Solving the system of equations:

1. From the first equation:
\[
-11 = -2a + 5b
\]
Simplify:
\[
2a - 5b = 11 \quad \text{(Equation 1)}
\]

2. From the second equation:
\[
9 = -3a + 5b
\]
Simplify:
\[
3a - 5b = -9 \quad \text{(Equation 2)}
\]

Now, subtract Equation 2 from Equation 1 to eliminate \( b \):

\[
(2a - 5b) - (3a - 5b) = 11 - (-9)
\]
\[
2a - 5b - 3a + 5b = 11 + 9
\]
\[
- a = 20
\]
\[
a = -20
\]

Substitute \( a = -20 \) into Equation 1:

\[
2(-20) - 5b = 11
\]
\[
-40 - 5b = 11
\]
\[
-5b = 51
\]
\[
b = -\frac{51}{5}
\]

Thus, the scalars are:
\[
a = -20, \quad b = -\frac{51}{5}
\]

Therefore, the vector \( \mathbf{v} = \langle -11, 9 \rangle \) can be expressed as a linear combination:
\[
\mathbf{v} = -20 \mathbf{x} - \frac{51}{5} \mathbf{y}
\]

Would you like more details or further clarification?

Here are 5 related questions:
1. How can you interpret the geometric meaning of this linear combination?
2. What does the solution tell us about the relationship between the vectors \( \mathbf{x} \) and \( \mathbf{y} \)?
3. How would the process change if the vectors \( \mathbf{x} \) and \( \mathbf{y} \) were linearly dependent?
4. Can this method be extended to 3-dimensional vectors?
5. How can this linear combination be visualized graphically?

**Tip:** To check if a linear combination is correct, substitute the values of the scalars into the original equation and verify both components match!

Express the vector v = <-11 , 9> as a linear combination of x = <-2, -3> and y = <5, 5>

Learn how to express the vector v = <-11, 9> as a linear combination of the vectors x = <-2, -3> and y = <5, 5>. This detailed solution involves solving a system of linear equations and explores key concepts in linear algebra, such as vector spaces and linear independence. Suitable for students in Grades 10-12 and college-level math courses.

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